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Find the equation of the line perpendicular to \(\displaystyle y = \frac{5 x}{9} - 8\) that contains \(\displaystyle \left( 2, \ 7\right)\).

The slope of the line is \(\displaystyle m = \frac{5}{9}\) so the perpendicular line has slope \(\displaystyle m=- \frac{9}{5}\). The equation has the form \(\displaystyle y=- \frac{9}{5}x+b\). Using the point \(\displaystyle \left( 2, \ 7\right)\) gives the equation \(\displaystyle 7=- \frac{9}{5}\left(2\right)+b\) Solving for \(\displaystyle b\) gives \(\displaystyle b = \frac{53}{5}\). The equation of the perpendicular line is \(\displaystyle y = \frac{53}{5} - \frac{9 x}{5}\).

Download \(\LaTeX\) 

\begin{question}Find the equation of the line perpendicular to $y = \frac{5 x}{9} - 8$ that contains $\left( 2, \  7\right)$. 
    \soln{9cm}{The slope of the line is $m = \frac{5}{9}$ so the perpendicular line has slope $m=- \frac{9}{5}$. The equation has the form $y=- \frac{9}{5}x+b$. Using the point $\left( 2, \  7\right)$ gives the equation $7=- \frac{9}{5}\left(2\right)+b$ Solving for $b$ gives $b = \frac{53}{5}$.  The equation of the perpendicular line is $y = \frac{53}{5} - \frac{9 x}{5}$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Find the equation of the line perpendicular to  <img class="equation_image" title=" \displaystyle y = \frac{5 x}{9} - 8 " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B5%20x%7D%7B9%7D%20-%208%20" alt="LaTeX:  \displaystyle y = \frac{5 x}{9} - 8 " data-equation-content=" \displaystyle y = \frac{5 x}{9} - 8 " />  that contains  <img class="equation_image" title=" \displaystyle \left( 2, \  7\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%202%2C%20%5C%20%207%5Cright%29%20" alt="LaTeX:  \displaystyle \left( 2, \  7\right) " data-equation-content=" \displaystyle \left( 2, \  7\right) " /> . </p> </p>
HTML for Canvas
<p> <p>The slope of the line is  <img class="equation_image" title=" \displaystyle m = \frac{5}{9} " src="/equation_images/%20%5Cdisplaystyle%20m%20%3D%20%5Cfrac%7B5%7D%7B9%7D%20" alt="LaTeX:  \displaystyle m = \frac{5}{9} " data-equation-content=" \displaystyle m = \frac{5}{9} " />  so the perpendicular line has slope  <img class="equation_image" title=" \displaystyle m=- \frac{9}{5} " src="/equation_images/%20%5Cdisplaystyle%20m%3D-%20%5Cfrac%7B9%7D%7B5%7D%20" alt="LaTeX:  \displaystyle m=- \frac{9}{5} " data-equation-content=" \displaystyle m=- \frac{9}{5} " /> . The equation has the form  <img class="equation_image" title=" \displaystyle y=- \frac{9}{5}x+b " src="/equation_images/%20%5Cdisplaystyle%20y%3D-%20%5Cfrac%7B9%7D%7B5%7Dx%2Bb%20" alt="LaTeX:  \displaystyle y=- \frac{9}{5}x+b " data-equation-content=" \displaystyle y=- \frac{9}{5}x+b " /> . Using the point  <img class="equation_image" title=" \displaystyle \left( 2, \  7\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%202%2C%20%5C%20%207%5Cright%29%20" alt="LaTeX:  \displaystyle \left( 2, \  7\right) " data-equation-content=" \displaystyle \left( 2, \  7\right) " />  gives the equation  <img class="equation_image" title=" \displaystyle 7=- \frac{9}{5}\left(2\right)+b " src="/equation_images/%20%5Cdisplaystyle%207%3D-%20%5Cfrac%7B9%7D%7B5%7D%5Cleft%282%5Cright%29%2Bb%20" alt="LaTeX:  \displaystyle 7=- \frac{9}{5}\left(2\right)+b " data-equation-content=" \displaystyle 7=- \frac{9}{5}\left(2\right)+b " />  Solving for  <img class="equation_image" title=" \displaystyle b " src="/equation_images/%20%5Cdisplaystyle%20b%20" alt="LaTeX:  \displaystyle b " data-equation-content=" \displaystyle b " />  gives  <img class="equation_image" title=" \displaystyle b = \frac{53}{5} " src="/equation_images/%20%5Cdisplaystyle%20b%20%3D%20%5Cfrac%7B53%7D%7B5%7D%20" alt="LaTeX:  \displaystyle b = \frac{53}{5} " data-equation-content=" \displaystyle b = \frac{53}{5} " /> .  The equation of the perpendicular line is  <img class="equation_image" title=" \displaystyle y = \frac{53}{5} - \frac{9 x}{5} " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B53%7D%7B5%7D%20-%20%5Cfrac%7B9%20x%7D%7B5%7D%20" alt="LaTeX:  \displaystyle y = \frac{53}{5} - \frac{9 x}{5} " data-equation-content=" \displaystyle y = \frac{53}{5} - \frac{9 x}{5} " /> . </p> </p>